Eulers Totient Function کا حساب کیسے لگائیں
Eulers Totient Function کیا ہے؟
Euler's totient function φ(n) counts how many integers from 1 to n are coprime to n (share no common factor other than 1). It is fundamental in number theory and RSA encryption.
فارمولا
φ(n) = n × ∏(1 − 1/p) for all prime factors p of n; for prime p: φ(p) = p−1
- n
- positive integer
- φ(n)
- Euler totient of n — count of integers coprime to n
مرحلہ وار گائیڈ
- 1For prime p: φ(p) = p−1
- 2φ(pᵏ) = pᵏ−pᵏ⁻¹
- 3Multiplicative: φ(mn) = φ(m)φ(n) when gcd(m,n)=1
- 4φ(12) = φ(4)×φ(3) = 2×2 = 4
حل شدہ مثالیں
ان پٹ
φ(12)
نتیجہ
4 (coprime: 1,5,7,11)
ان پٹ
φ(7)
نتیجہ
6 (prime: all 1–6 are coprime)
اکثر پوچھے جانے والے سوالات
Why is φ(n) important in cryptography?
φ(n) is essential to RSA encryption: the security depends on the difficulty of computing φ for large products of primes.
What does "coprime" mean?
Two numbers are coprime if their greatest common divisor (GCD) is 1. They share no common factor except 1.
What is φ(p) for a prime p?
φ(p) = p−1, because all numbers 1 to p−1 are coprime to p.
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